American Institute of Mathematical Sciences

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June  2007, 7(4): 779-792. doi: 10.3934/dcdsb.2007.7.779

Circular and elliptic orbits in a feedback-mediated chemostat

Willard S. Keeran 1, , Patrick D. Leenheer 1, and  Sergei S. Pilyugin 1,

1. 

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, United States, United States, United States

Received  August 2006 Revised  January 2007 Published  March 2007

A chemostat with two organisms competing for a single growth-limiting nutrient controlled by feedback-mediated dilution rate is analyzed. A specific feedback function is constructed which yields circular and elliptical periodic orbits for the limiting system. A theorem on the stabilization of periodic orbits in conservative systems is developed and for a given elliptical orbit, the result is used to modify the chemostat so that the chosen orbit is asymptotically stable. Finally, the feedback function is modified so that finitely many nested periodic orbits of alternating stability exist.
Keywords: conservative system., feedback-mediated control, periodic coexistence, Chemostat.
Mathematics Subject Classification: 92D25, 92D37, 92D4.
Citation: Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Circular and elliptic orbits in a feedback-mediated chemostat. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 779-792. doi: 10.3934/dcdsb.2007.7.779
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